Lecture Plan for Mathematics 4N
The following plan is preliminary and subject to change.
For the analytical parts of the lecture, we will mostly follow the book Advanced Engineering Mathematics by Erwin Kreyszig, 10th edition, John Wiley & Sons, 2011, and the chapter numbers in the table below refer to this textbook. The numerical parts are based on our own notes; we will publish them during the course of the semester.
Estimated Weekly lecture plan
For more corresponding Jupyter notebooks see also the Exercises. The additional PDFs we provide in the Literature column sometimes refer to example python code. This is included in the Python code accompanying the lecture, because the examples in the slides agree with the ones in the additional Literature. For that reason we do not publish the code twice. If you miss something specifically, please open a post in the forum.
| Week | Topic | Additional material | Literature | |
|---|---|---|---|---|
| Numerical mathematics | ||||
| 2 | Introduction and Numerical mathematics Example: chemical reactor Two-point boundary value problems Numerical differentiation | Slides: 01_preliminaries-bvp-web.pdf (V2, 14/01/2025, fixed typos) (Jupyter-)notebook: 01-preliminaries-bvp.ipynb | Preliminaries: preliminaries.pdf (V2, 16/01/2025, fixed a typo in Ex4) or preliminaries.ipynb boundary value problems: bvp.pdf (V2, 17/01/2025, fixes a sign error in Ex2) | |
| 3 | Polynomial interpolation Spline interpolation Numerical integration Simple and composite quadrature rules Adaptive integration | Slides: 02-interpolation-quadrature_web.pdf Notebook: 02-interpolation-quadrature.ipynb | Interpolation: interpolation.pdf (Sec. 1, Sec. 2 – only direct approach, Sec. 3) Numerical Integration: quadrature.pdf Kreyszig, Ch. 19.3-19.5 | |
| 4 | Python with numpy Rounding errors Numerical solution of nonlinear equations Fixed-point iterations | Slides: 03-rounding-errors-nonlinear-equations-web.pdf Notebooks: 03-numerical-errors.ipynb, 04-numerical-solution-of-nonlinear-equations.ipynb | Nonlinear Equations: nonlinearequations.pdf Kreyszig, Ch. 19.1 | |
| 5 | Newton's method Newton's method in several variables | Slides: 04-newton-web.pdf Notebook: 04-numerical-solution-of-nonlinear-equations.ipynb (cont. from last week) | Nonlinear Equations: nonlinearequations.pdf Kreyszig Ch. 19.2 | |
| Fourier analysis | ||||
| 6 | Periodic functions Euler's formula (complex) Fourier series Conv. of Fourier series Changes of the period Half-range expansions Approx. with trig. pol. | Slides: 05-fourier-series_web.pdf (V2, 27/03) Notebook: 05-fourier-series.ipynb Computation of the Fourier coefficients for the piecewise constant function on slide 5-26: 05-fourier-notes-side-5-26.pdf | Kreyszig, Ch. 11.1, 11.2, 11.4 additionally: Plonka, Potts, Steidl, Tasche: Numerical Fourier Analysis, Birkhäuser, 2nd Edition, 2023 (PDF freely available from NTNU). Chapter 1 for a general introduction to Fourier Series * 1.2 for Fourier coefficients * 1.3 for convolution * 1.4 for convergenc, especially 1.4.3 for the Gibbs Phenomenon | |
| 7 | Fourier transform Properties of the Fourier transform Convolution Discrete Fourier Transform (DFT) | Slides: 06-fourier-transforms-web.pdf Notebook for DFT: 06-discrete-fourier-transform.ipynb | Kreyszig Ch. 11.9 additionally: Plonka, Potts, Steidl, Tasche: Numerical Fourier Analysis, Birkhäuser, 2nd Edition, 2023 (PDF freely available from NTNU). * Chapter 2 for more details on the Fourier transform * Chapter 3 for the Discrete Fourier Transform (DFT), see also Section 1.5 * Chapter 5 if you want to read more about the FFT | |
| Partial differential equations | ||||
| 8 | Partial differential equations (PDEs) Wave equation Separations of variables | Slides: 07-partial-differential-equations-web.pdf (V2, 16/04) Notebook: Pluto-notebook as a pdf A Desmos Sketch of the same example: String example | Kreyszig, Ch. 12.1, 12.2, 12.3 | |
| 9 | d'Alembert's solution method Numerics for the wave equation Heat equation | Slides: 08-numerics-for-pdes-web.pdf (last version 22/04) Notebook I: 08-numerical-methods-pdes-i-wave-equation.ipynb Notebook II: 08-numerical-methods-pdes-ii-heat-equation.ipynb Video of the Tacoma bridge collapse | Numerical solution to PDEs: numpde.pdf (V2, 22/04) Kreyszig, Ch. 12.4, 12.6 | |
| 10 | Numerics of the heat equation Fourier methods for PDEs | Kreyszig, Ch. 12.7 | ||
| Laplace transform | ||||
| 11 | Laplace transform | Slides: 09-laplace-transform-web.pdf | Kreyszig, Ch. 6.1, 6.2 | |
| 12 | Laplace transform | Kreyszig, Ch. 6.3, 6.4, 6.5, 6.7 | ||
| Numerics of ordinary differential equations | ||||
| 13 | Numerics of initial value problems | Slides: 10-numerics-for-odes-web.pdf (V2, 03/04/25) Notebooks: 10a-numerical-methods-for-odes.ipynb, 10b-adaptivity.ipynb | Numerical solution to ODEs: numode.pdf | |
| 14 | Numerics of initial value problems | Numerical solution to ODEs (Part II): numode_part2.pdf | ||
| Summary and revision | ||||
| 15 | Summary and Exam practice | 11-summary-web.pdf (V2, 09/04) | ||
| 16 | Easter | |||
For numerics we will heavily use Jupyter Notebooks (.ipynb-files). There is two ways you can work with these.
Jupyterhub
The fastest way to get started with Python and the Jupyter ecosystem is to use our Jupyterhub, which provides a cloud-based Jupyter environment which you can simply use in your browser. Just click on the link, enter your NTNU/Feide credantials and voilà, a Jupyter notebook application will show up in your browser tab.
Anaconda
If you want to work on your computer and not in the browser, we recommend using the Anaconda distribution, a detailed installation guide for which can be found here.
In order to work with jupyter notebooks in Anaconda, follow the following steps:
- (Download and install Anaconda on your computer.)
- Download the jupyter notebook on your computer. We recommend to set up a dedicated working directory for Maths 4N/D.
- Start the Anaconda Navigator, and launch Jupyter Notebook. This should open a new window in your web-browser, from which you can navigate to your stored notebooks or open new ones.